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![](http://upload.wikimedia.org/wikipedia/commons/thumb/9/95/Lattice_of_the_divisors_of_60%2C_ordered_by_divisibility%3B_with_divisors_of_30_in_red.svg/220px-Lattice_of_the_divisors_of_60%2C_ordered_by_divisibility%3B_with_divisors_of_30_in_red.svg.png)
In mathematics, especially in order theory, the greatest element of a subset of a partially ordered set (poset) is an element of that is greater than every other element of . The term least element is defined dually, that is, it is an element of that is smaller than every other element of
Definitions
Let be a preordered set and let An element is said to be a greatest element of if and if it also satisfies:
- for all
By switching the side of the relation that is on in the above definition, the definition of a least element of is obtained. Explicitly, an element is said to be a least element of if and if it also satisfies:
- for all
If is also a partially ordered set then can have at most one greatest element and it can have at most one least element. Whenever a greatest element of exists and is unique then this element is called the greatest element of . The terminology the least element of is defined similarly.
If has a greatest element (resp. a least element) then this element is also called a top (resp. a bottom) of
Relationship to upper/lower bounds
Greatest elements are closely related to upper bounds.
Let be a preordered set and let An upper bound of in is an element such that and for all Importantly, an upper bound of in is not required to be an element of
If then is a greatest element of if and only if is an upper bound of in and In particular, any greatest element of is also an upper bound of (in ) but an upper bound of in is a greatest element of if and only if it belongs to In the particular case where the definition of " is an upper bound of in " becomes: is an element such that and for all which is completely identical to the definition of a greatest element given before. Thus is a greatest element of if and only if is an upper bound of in .
If
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